When attempting to derive quantum theory, one of the biggest questions that we have to answer is: why the Hilbert space? Replacing probability distributions with wave functions gives rise to the characteristic features of quantum theory, such as superpositions, entanglement, and non-commuting operators. To answer the question of how Hilbert spaces and operator algebras arise in quantum theory, researchers in quantum foundations often concentrate on other, more operational aspects of quantum theory: on the convexity of sets of density matrices (or sets of states in general) and on compositionality of the operations we can carry out in a laboratory. These properties give rise to different, yet similar frameworks: general probabilistic theories [1], operational probabilistic theories [2], and effectus theory [3] to name a few. In all of these frameworks, one can not only construct models of classical and quantum theory, but also other more exotic examples of hypothetical theories, such as the “Boxworld” theory [4] that contains the Popescu-Rohrlich (PR) boxes [5].

Effect algebras and sequential product

Convex effect algebras (or, equivalently, ordered vector spaces) usually play a key role in all of the aforementioned frameworks used in foundations of quantum theory. Effect algebras describe the set of all possible two-outcome (i.e., “yes”-“no”) measurements that one can perform in a hypothetical theory. Effect algebras were introduced in 1994 by Foulis and Bennet [6], but an equivalent framework of D-posets was introduced in 1992 by Kôpka [7]. Since then, effect algebras have grown into a research field of their own. There are several foundational aspects of effect algebras that one can investigate: apart from convexity (not every effect algebra has to be convex), spectrality and spectral decompositions were investigated in the framework of effect algebras [8], since many calculations in quantum theory rely on spectral decompositions of operators into projectors and the spectral theorem.

In a paper [9] recently published in Quantum, A. Westerbaan, B. Westerbaan and J. van de Wetering have investigated another property of effect algebras: the compositionality of effects and sequential products. An effect is an element of an effect algebra and it describes a single measurement on a (physical or hypothetical) system with a “yes” or “no” answer. Given a state of the system, an effect gives us the probability $p$ of obtaining the “yes” answer if we perform the corresponding measurement, the probability of obtaining the “no” answer is $1-p$, given by the normalization. A natural question arises: how do we describe a situation when we first perform a measurement corresponding to an effect $a$ and then we perform a subsequent measurement corresponding to the effect $b$ on the same system? This is a very basic scenario that one can surely implement in any laboratory.

One can either employ the Schrödinger picture and describe the change of the state of the system when performing the measurement corresponding to $a$, or one can employ the Heisenberg picture and describe a transformation of the effect $b$ that is given by the collapse of the state during measurement corresponding to the effect $a$; in quantum theory this is described by the Lüders product that maps $b \mapsto \sqrt{a} b \sqrt{a}$. One uses the Heisenberg picture when describing everything in terms of effect algebras.

In a general effect algebra, we do not have access to the square root and we can not multiply effects, so we can not directly generalize the Lüders product. But we can define the sequential product $\&$ (also denoted $\circ$) such that $a \& b$ (or $a \circ b$) is the effect that replaces $b$ when we first measure $a$, e.g., in quantum theory $a \& b = \sqrt{a} b \sqrt{a}$. This gives rise to the concept of a sequential effect algebra, or SEA for short. And this is the starting point for A. Westerbaan, B. Westerbaan and J. van de Wetering in [9].

The result: sequential product (almost) implies convexity

The result of [9] is characterized by the equation
$$
E = E_b \oplus E_c \oplus E_{ac},
$$
which says that any normal SEA $E$ can be written as a sum of three distinct effect algebras, each with different properties and interpretation.

• $E_b$ is a Boolean algebra and it represents a classical, deterministic system. The Boolean algebra represents the standard logic of classical computers: it does not allow for the creation of randomness and the outcome of every possible measurement is predetermined and either always occurs with probability $p=1$ or it never occurs, i.e., $p=0$.
• $E_c$ is a normal convex SEA. This is the type of effect algebra that we encounter in quantum theory. Also, $E_c$ is a convex effect algebra, such as the ones we encounter in almost every framework used within quantum foundations.
• $E_{ac}$ is purely almost-convex SEA. This is a new type of normal SEA, that was identified by the authors, and that they deem pathological. It is of future interest, whether purely almost-convex SEAs are pathological and whether we should use axioms as the ones presented by the authors to eliminate purely almost-convex SEAs, or, whether we should include them into our frameworks.

The result of A. Westerbaan, B. Westerbaan and J. van de Wetering is similar in spirit to the famous characterization of factors of von Neumann algebras [10] and it has the potential to be as impactful on the foundations of quantum theory. This is because the result shows that one can (almost) derive convexity of the effect algebra from the sequential product. It follows that one can replace convexity as a basic assumption with the sequential structure of the effect algebra. This result, together with the earlier result that one can derive quantum theory from a continuous sequential product [11], showcase the power and the potential of sequential products.

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